Theory of complete orthogonal direct decompositions of vector spaces

A theory is constructed for the complete orthogonal (with respect to generalized orthogonalities defined by certain partial symmetric bilinear functions) direct decompositions of vector spaces $V$ such that the quotient of $V$ by a certain particular subspace is finite-dimensional. The main result of the theory is a description of all such decompositions. This theory has an application to the theory of direct decompositions of $p$-ary functions, where $p$ is a prime.